Math Wars: The Debate Between Higher-Order Vs. Rote Learning

Recently, EdSurge published a fabulous post highlighting the escalating rhetoric that the Khan Academy has inspired among math educators and edupreneurs. Sal Khan’s success has brought to the forefront a discussion that has been ongoing in academic and education circles for some time. This debate parallels the one about Common Core Math Standards exemplified by the Wurman and Wilson article referenced in a recent Getting Smart post.

At the heart of the debates is the tension between teaching students to accurately perform math computation and procedures versus teaching students higher-order mathematics skills. Versions of this debate have persisted through numerous iterations of math reform. As early as 1965, Tom Lehrer quipped in his song, New Math, “but in the new approach, as you know, the important thing is to understand what you’re doing rather than to get the right answer,” a perspective that summarizes the skepticism of parents and employers and leads many an edupreneur to focus on the “rote skills” of memorizing number facts and solving problems procedurally.

And yet, there seems to be general agreement that we want kids to understand the math as well as be able to perform it. Traditional thought suggests that before students can learn higher-order thinking about math, they need to be fluent in the basics. It turns out, though, that there are counterexamples. Without knowing any math facts even very young students can accurately perform and gain a conceptual understanding of division through equipartitioning, for example. Further, it turns out that with a higher-order framework in place, the process of recalling the associated facts and procedures becomes more efficient thanks to the way the brain chunks them together into meaningful units as it develops expertise. I’m no mathematician or researcher, but as a mom this seems to me to imply that I should strive to offer my children opportunities to gain mathematical understanding before asking them to become efficient problem-solvers and computational whizzes.

But then, that same research points out that the chunking that happens also relies on deliberate practice – about 10 years of it to become an expert in anything. So perhaps I should instead insist that my kids gain fluency in rote computation so that as they tackle higher-order math, whether applying computation in real-world situations or solving algebraic equations, they are not spending excessive cognitive energy on solving the 3d grade part of the problem. Perhaps like katas in martial arts, the deliberate practice to automaticity of rote mathematical facts and procedures provides the tools necessary to grapple with real mathematical problems – a process which I don’t doubt is necessary to gain authentic higher-order skills.

Higher-Order Thinking Versus Rote Learning

Separating out the learning of conceptual understanding from procedural fluency is non-trivial. One challenge, as I see it, is to find ways of instilling higher-order skills without turning them into just another layer of abstraction in rote procedural learning.

Here’s an example of that problem: If a student can quickly fill out an addition worksheet, that doesn’t provide much information regarding whether she understands addition conceptually. To see whether she really understands addition, we may want to ask her a real-world question so we construct a word problem along the lines of “Jane has 11 apples and Molly has 6. How many do they have altogether?” Our student may struggle with this word problem so in order for her to pass the new test, we teach her how to read the word ‘altogether’ as a clue to recognize that this is an addition problem. We wanted to know if a student could add, so we gave her addition problems. It turns out she could learn those by rote so we asked a higher-order question in the form of a word problem. But the “teach to the test” arms race then created rote learning for solving that type of problem, too.

How do you develop assessments of student understanding that can be tested en masse but is not subject to proceduralization? And if the goal is to avoid proceduralization, how do you develop digital tools to support higher-level conceptual learning? Is the computer truly not a natural medium for mathematical learning? And if so, does that also rule out the computer as a mechanism for connecting communities of learning that might better provide that personal, human support? Or do these questions simply demonstrate a failure of the imagination?

So what do students need from us in order to develop both mathematical fluency and higher-order mathematical thinking? Speaking again as a mom of two very different children, and not as an education scientist or researcher, I suspect we need authentic problems to work on and opportunities to practice skills until they become deeply ingrained, something that looks more like on-the-job training than leture-and-test. If this is, indeed, true, the Khan Academy debate will need to be reframed.

Khan Academy Raises the Floor

Which brings me to what I love about Khan Academy – it raises the floor. Although it doesn’t teach higher-order math skills, neither do many math teachers. In the public school system you get the teacher you get, generally without appeal, and most likely you will receive traditional lecture and test pedagogy at a set pace. If you are lucky, you will get more enlightened math education, yet the odds are good that you will still end up performing rote procedures…just at a higher level of abstraction. If you are extremely lucky and learn math from a mathematician, you are still one of 30 kids being guided through the beauty that is math in lockstep. But in the classrooms that are using the Khan academy, students:

Have more autonomy – not only can students move at their own pace, the entire learning map is spread out in front of them and they can move around it according to their own choices

Have the experience of achieving mastery – mastering Khan (that is, the specific subset of things it offers) is very straightforward with charts, badges, and reports that give the student a clear sense of progress and of distance remaining

Have fun – Sal Khan is a pretty entertaining guy, not a bad choice if you are destined to learn by lecture anyway

And in some cases, gain a sense of purpose – in many classrooms, educators are experimenting with using class time to explore authentic questions and problems and letting students watch Khan’s videos at home to find the tools they need to address those questions.

I was prepared, based on my biases and theories and parenting experience to disdain Khan before it was cool to disdain Khan. But here’s the thing: I tried it and found it fun, inviting, and engaging. The badging design is smart and understated so my kids want to collect the badges. The problem solving is somewhat adaptive so no one has to spend hours on rote work they’ve already mastered. When combined with the support of a good teacher, it reduces much of the friction associated with learning traditional materials in traditional ways. Most importantly, it can be extremely efficient in covering the “basics” leaving far more time and space for the higher-order problem solving, open exploration, collaborative work, and project-based learning that so often gets squeezed out of classrooms.

As much as I would LOVE to use a tool, technique, or approach that would help my kids achieve computational and procedural fluency as a side effect of authentic problem solving, to date I have found those only in theory and aspirations. I therefore welcome the tools, limited though they may be, that make that process even the tiniest bit easier. Khan Academy does this, raising the floor in a way that is intrinsically scalable and affordable.

Now it’s up to the education and entrepreneurial communities together to raise the ceiling. Higher-order problem solving and procedural automaticity are inextricably intertwined. Khan’s contribution to a subset of that space is far less a barrier to the development of truly innovative on-line mathematical learning tools than the prevailing mindset of least-common-denominator rote testing, and the collective memory of the adult community of what school is supposed to be like. It’s not that Khan Academy is close to all I would hope for – it’s just better than the alternatives for many, many people. More importantly, it still looks like math to the traditional community of parents, teachers, entrepreneurs and employers while starting to show just some of what is possible when digital tools help make education more personalized.

At the end of their blog post, EdSurge calls for both approaches to be reconciled – I second that and call for a vote. What do you think?

This means each mathematical concept builds on the previous. Wholistic or Higher learning, cannot come first because there would be no information to draw from to form a good understanding of a higher level concept.

I believe H. Wu is right about fundamentals needing to be there before a wholistic understanding is possible. see below link to his writing.

An analogy might be to compare “higher level or wholistic ideas when teaching math without fundamental practice” to that of explaining to a young athlete the strategies of how to win a game and a coach sending them out on the court, or field or ice rink with a description of how to win a game, but with no or very little skills or practice of fundamentals. THe younger the athlete is when they get the fundamentals down, (and it doesnt come from doing someithing once)- the better athlete they will become. Good fundamentals need to be taught very early on before bad habits are developed, or the athlete will never perform up to their potential. The best time to teach skills is around 9 or 10- due to the timing of how and when the brain is forming learning paths. Fundamentals and practice of those fundaments until they become rudamentary are a necessary part of winning the game. And the best athletes have all of the reasoning skills, the ball sense, knowing what position to be in, understanding you need to pass, but without fundamental skill, all is lost. It would be ludicris to try and teach how to win at sports with no fundamental training previous. But that is how the concept of mathematics is being taught in many curriculums around the country. We have one of the worst in our school district- Connected Math. The result is massive confusion among the students, very upset parents, who then seek charter or private schools -who offer, hopefully, a better approach ( some do, some don’t) But I see curriculums as being very poorly structured and nonsequential. I see the Khan academy as being a partial answer and a great way to explain a fundamental skill. The students themselves have to do the practicing of the skills, and this needs to be provided via a curriculum with daily requirements or weekly requirements for practice. At the same time, larger explanations can be incorporated, but they cannot stand alone.

Math is a subject that requires understanding and reasoning, something that needs basic analytic ability to derive meaning out of seemingly cumbersome tasks, yet it requires a lot of memorization of formulas and procedures. So to say that only higher order or only rote learning is the way to go will be stepping on one’s own feet. The fact is we have to come up with a pedagogy that justifies and blends both. For that you need to first start with great content, which is interactively put in a simple engaging style, something that CK12 and Wolfram have done exceedingly well with their Enhanced Algebra I, http://www.ck12.org/about/ck12-wolfram-algebra. And as far as Khan academy is concerned, well its impact on the performance of students in maths is undoubtedly debatable, but what it has done is turned staid lectures into fun simple sessions…and that’s about it!

Nice post. I like the way you phrase KA’s contribution as raising the floor. KA is the natural extension of the traditional model of instruction. If a classroom teacher cannot provide something more for his or her students than KA, he or she is not doing her job. Unfortunately, as you mention, many teachers are not–hence the ire against KA.

The other unfortunate aspect of the KA controversy is that KA does not solve the problem or move us toward a solution. The advantages you list are true, but they don’t solve the problem, it is simply a slight improvement on a failing model.

@Gretchen: In response to the idea that learning math is like learning an athletic skill, your analogy is not at all appropriate. The brain is not a muscle and thus should not be trained like a muscle. Research continues to support this idea and has for over a century. Even the work of Thorndike (who proposed the Law of Exercise–the idea that you are referring too) admitted that the practice must be meaningful, and he followed up the law of exercise with the Law of Effect which emphasizes the importance of motivation in realizing the benefits of practice. Even Plato knew this saying: “Bodily exercise, when compulsory, does no harm to the body;
but knowledge which is acquired under compulsion obtains no hold on the mind.”

You disparage the Connected Math curriculum, but do you have any data to support your claim that it is failing? I suspect that you do not, because many studies have shown that this curriculum, and nearly all of the other reform curricula of the 90’s are just as effective as their traditional counterparts in terms of rote skills, and tend to exceed the traditional curricula in terms of comprehension and non-routine problem solving ability.

@Aran – I agree the model needs to change. Still KA does solve a very particular and pragmatic problem, that of helping my own kids with learning today. And I very much hope that it, along with others, can point the way to new models.

Also, thanks for noting the “raising the floor” metaphor. I’ve gotten so much great feedback on that now I have to publicly give the credit to Tom Vander Ark’s book, Getting Smart, where the metaphor originated. I’m merely citing examples in support – as you can tell, I like it, too.

For a problem solving based program that works brilliantly for my children, try artofproblemsolving.com. I haven’t tried khan for years, but back when it started it was dead dull, and frustratingly slow…

regarding the brain training…it’s exactly like athletic training. No, the brain is not a muscle, but it is trained by repetition over time, the same way you learn any skill (athletic, piano, ride a bike) etc… it’s not just the body…it’s the brain AND the body… training is training, and basics are necessary before you can build intermediate/advanced skills

Also, evidence of “Integrated Math” failing? Absolutely. Right here in GA. Done. It was a failure…but I could’ve told you that before they got started. Mathematics are ‘discrete’ branches…thus, can’t really be combined. Or, combining them to teach them is forced, unnatural and doesn’t fit the material…and won’t produce a good result.

But it’s missing the big picture: it’s not either/or, it’s both/and.

That is, like the article said, kids can tell how to divide a thing into two pieces…yes, they know what ‘division’ is… so that’s how you access them about the concept… then you show them how to write it in squiggles more efficiently (aka arabic numerals)… and you show them how to work with the numbers IN THE CONTEXT of apples, rectangles, cars, whatever that you’re counting.

and they spend time working with the material…so that they know it ‘by heart’… yes, they memorized it, yes they understand it…

Thus, they are fluent… and the current research shows that arithmetic fluency has strong correlation with PSAT scores, self-confidence, and higher mathematics… Can’t run until you can walk…

and practically… I just gave an arithmetic 100 question 2-minute test (adding, subtracting, multiplying… single-digit) to some High School Seniors who failed their Math Graduation test twice already… and none of them passed it. I actually gave them three minutes and only two completed it (2:45) the others got about half-way through

So, how do they expect to solve higher level problems if they can’t add 13 +7? if they can’t do their times tables without their fingers?

That’s what the data (neuro-psychological, educational, etc) are showing…

(not sure if anyone will read this, since it’s a year later…but it feels good to say it anyway 🙂